Chapter 6: Graphical presentation of causal effects

Causal directed acyclic graphs

  1. Lack of arrow indicates absence of causal effect
  2. All causes, including unobserved, are in the DAG
  3. Any variable is a cause of its decedents

\[ Y \perp Z \mid X, W \]

Defining property: “… conditional on its direct causes, any variable on the DAG is independent of any other variable for which it is not a cause.”

DAGs and counterfactuals

  • Variables in DAGs are variables on which one can intervene, they can be set to specific values, i.e. to all potential realizations of a variable
  • That is, one ca set variables to possibly counterfactual, values, e.g. \(x=0\) or \(x=1\), which are used to describe potential outcomes \(Y^0\) or \(Y^1\)
  • In nonparametetric structural equation models (SEM) we define “counterfactual worlds” by setting the values of the exogenous variables (to which no arrow leads) and & setting the values of children-variables given the values of the parents and the parent-child functions.

Causal diagrams and marginal independence

  • DAGs capture causation and association between variable \(V\)
  • A Path “connects variables by following a sequence of edges such that the route visits each \(V\) no more than once”
  • causal path: has an arrow leading in & out of every \(V\)
  • association path: can include \(V\)s with only out-going edges.

Flows of marginal association

Fork: When only S causes B and D, B and D will be correlated

Mediator: When X causes Y via M, X and Y will be correlated

Collider: When K is caused by independent X and Y, X and Y are uncorrelated

Figure 1: Association flows along all open paths

“Two variables are marginally associated if one causes the other, or if they have a common cause.”

Causal vs associative relationship

  • Causal relationship:
    \(\small Pr[Y^{a=1}=1] - Pr[Y^{a=0}=1]\)
  • Associative relationship: \(\small Pr[Y=1|A=1] - Pr[Y=1|A=0]\)

The difference between observed conditional probabilities is not the same as the difference between two counterfactuals.

\[ \tiny Pr[Y^{a=1}=1] - Pr[Y^{a=0}=1] \neq Pr[Y=1|A=1] - Pr[Y=1|A=0] \]

Causal diagrams and conditional independence

Does the association between two variables depend on the value of a third variable?

Setting a variable to a value is the same as conditioning on that variable. Conditioning can open or close paths.

Flows of conditional association

Conditioned fork: When only S causes B and D and we fix S to a specific value, B and D will be uncorrelated

Conditioned mediator: When X causes Y via M and we fix M to a valye, X and Y will be uncorrelated

Conditioned collider: When B is caused by independent E and I but we fix B, E and I are correlated due to their common effect.

Figure 2: Association flows along all open paths

d-separation1

A path is blocked if and only if it contains a non-collider that has been conditioned on, or it contains a collider that has not been conditioned on and has no descendants that have been conditioned on. Two variables are d-separated if all paths between them are blocked (otherwise they are d-connected).

Conditioniong and biases

Conditioning on a …

  • … common cause1 of treatment & outcome removes confounding bias2
  • … mediator1 between treatment & outcome introduces bias
  • … collider1 between treatment & outcome introduces selection bias2

Positvity in causal diagrams

Positivity is roughly translated into graph language as the condition that the arroes from causes of treatment \(L\) to treatment \(A\) are not deterministic

\[ a = f(l) + \epsilon \]

Consistency in causal diagrams

  • Well defined interventions: \(A \rightarrow Y\) corresponds to an unambiguous, specific intervention

Treatment nodes are “special”, in that edges into them must bot be deterministic (positivity) and edges out of them towards the outcome must reflect a consistent treatment